Complex numbers can be represented in different ways:
 | i2 = j2 = -1 | ||||||||
| s = a + jb = r∟q | ||||||||
| conjugate of s = a - jb | ||||||||
| s = r*(cosq + j*sinq) | ||||||||
| where r = (a2 + b2) 1/2 and q = tan-1(b/a) | ||||||||
| s = rejq = r*(cosq + j*sinq) | ||||||||
| as = e sLn(a) | ||||||||
| if s1 = a1 + jb1
and s2 = a2 + jb2
|
If we have a transfer function defined in Laplace form we can convert it to the frequency domain simply by substituting jw for each occurrence of s in the transfer function
| 1 / (1 + s) = = 1 / ( 1 + jw) |
| 1 / (s2 + s + 1) = = 1 / (1 - w2 + jw) as s2 = ( jw)2 = -1*w2 |
The frequency response representation of complex numbers is a useful feature. For example sometimes we can only obtain the frequency response of a system and we need to convert this information to transfer function form to make further analysis easier.